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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Lawnmower

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

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What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Cyclic Quad Jigsaw

Age 14 to 16 Challenge Level:

Why do this problem?

First, the surprising amount of variation in possibilities shown in the video is worth the journey. Secondly, even though there is such variation the outer quadrilateral is always cyclic.Finally, specialising by trying numbers can help form a map of the journey you need to make in order to prove the generalisation for any cyclic quadrilaterals.

Possible approach

The first stage is simply to investigate:
Construct an image with the given constraints either using dynamic geometry or with a ruler and compasses. Using ruler and compasses is difficult simply because you need some flexibility to ensure a reasonable overlap of the circles.

Are learners surprised by the flexibility visible in the dynamic image?
Allow time for lots of discussion about construction techniques, the order of working (formed by the constraints) and the freedoms available (how many circles will meet the cirteria?).

Now for the problem.
A first step is to encourage exploration by writing in some angle sizes (following a discussion of the properties of opposite angles of a cyclic quadrilateral). Does the outside quadrilateral have opposite angles whose sum is 180 degrees and is therefore cyclic?

In specialising by using numbers for angles and keeping track of which angles can be calculated from others, the steps to a generalisation are much clearer.

Key questions

  • What defines a cyclic quadrilateral?
  • What are the freedoms?
  • What the contraints?

 

Possible support

Focus on the construction and looking at specific examples.
For work on cyclic quadrilaterals try Pegboard Quads.